Exercise 1 (algebraically closed fields). 1. Let K be an infinite field and F a subfield. Show that if K/F is algebraic and F infinite, then K and F have the same ...
Exercise 1 (algebraically closed fields) 1. Let K be an infinite field and F a subfield. Show that if K/F is algebraic and F infinite, then K and F have the same cardinality. Let B be a transcendence basis of K over F . Express |K| in terms of |B| and |F |. 2. Show that the Lring -theory of an algebraically closed field is not ℵ0 -categorical. 3. Show that the Lring -theory of an algebraically closed field is λ-categorical for every λ > ℵ0 . Exercise 2 (K-vector spaces) Let K be an infinite field and LK the language {+, mk , 0 : k ∈ K} where mk is unary function symbol for every k. A K-vector space V has a natural LK -structure where mk is interpreted putting mVk (x) = kx for every x in V . Let Σ be the theory of all vector-spaces. 1. Write down the axioms for Σ. 2. Show that Σ is not |K|-categorical. 3. Show that Σ is λ-categorical for every λ > |K|. Is Σ complete? 4. Does Σ eliminate quantifiers? Exercise 3 Let L be the language {=, cn : n ∈ N} where cn is a constant symbol for every n. Let N be the L-structure such that cN n = n, and let Σ be its theory. Let λ be an infinite cardinal. Is Σ λ-categorical? Does Σ eliminate quantifiers?
